#7563113

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2765 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(2,3)(6,7)(9,10,14,18)(11,15,22,30)(12,16,24,32)(13,17,26,34)(19,27,23,31)(20,28,21,29)(25,33,37,40)(35,38,36,39);(1,2,4,3)(6,7)(9,11)(10,15)(12,23)(13,20)(14,22)(16,31)(17,28)(18,30)(19,24)(21,26)(25,36)(27,32)(29,34)(33,39)(35,37)(38,40);(5,6,8,7)(9,12,14,24)(10,16,18,32)(11,19,22,23)(13,25,26,37)(15,27,30,31)(17,33,34,40)(20,35,21,36)(28,38,29,39);(5,7,8,6)(11,20)(13,26)(15,28)(17,34)(19,35)(21,22)(23,36)(25,37)(27,38)(29,30)(31,39)(33,40);(1,3,4,2)(12,24)(16,32)(19,23)(25,37)(27,31)(33,40)(35,36)(38,39)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7470309);